Are the following linear functions? Prove your conclusions...

Are the following linear functions? Prove your conclusions by showing that $f(\mathbf{r})$ satisfies both of equations ( $7.1$ ) or that it does not satisfy at least one of them.
$f(\mathbf{r})=\mathbf{A} \cdot(\mathbf{r}-\mathbf{k} z)$

Key Concepts

Linear Transformations

A linear transformation (or linear function) between vector spaces is one that preserves the operations of vector addition and scalar multiplication. This means that for any vectors u and v and any scalar c, the function must satisfy f(u + v) = f(u) + f(v) and f(cu) = c f(u). These requirements ensure that the transformation is consistent with the structure of the vector space, and are essential when proving or disproving the linearity of a given function.

Additivity and Homogeneity

These are the two fundamental properties that define a linear function. Additivity requires that the function's output for a sum of vectors equals the sum of the outputs for each vector individually, while homogeneity (or scalar multiplication) requires that scaling a vector and then applying the function is the same as applying the function and then scaling the result. Verifying these properties is crucial when assessing whether a function, such as one defined using a dot product with a shifted argument, is truly linear.

Zero Element Condition

An immediate consequence of the linearity properties is that the image of the zero vector must be the zero element of the codomain, i.e., f(0) = 0. This condition is vital because any deviation from it is a clear indicator that a function is not linear. For a function defined with a constant shift, if the output at the zero vector is nonzero, it violates linearity.

Dot Product

The dot product is a bilinear operation that is linear in each of its arguments separately. In the context of functions defined by a dot product, such as f(r) = A·(r ? k z), the linearity of the dot product with respect to the vector variable is used. However, if the function includes a constant term (such as A·(?k z)) that does not vanish when r is zero, this constant shift may disrupt the overall linearity of the function. Understanding the linearity properties of the dot product is therefore key in analyzing the linear nature of the function.

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